3. Let the random variables X and Y have the joint probability density function fxr (x,...
3. Let the random variables X and Y have the joint probability density function 0 y 1, 0 x < y fxy(x, y)y otherwise (a) Compute the joint expectation E(XY) (b) Compute the marginal expectations E(X) and E (Y) (c) Compute the covariance Cov(X, Y)
Let the random variable X and Y have the joint probability density function. fxy(x,y) lo, 3. Let the random variables X and Y have the joint probability density function fxy(x, y) = 0<y<1, 0<x<y otherwise (a) Compute the joint expectation E(XY). (b) Compute the marginal expectations E(X) and E(Y). (c) Compute the covariance Cov(X,Y).
2. Let X and Y be continuous random variables with joint probability density function fx,y(x,y) 0, otherwise (a) Compute the value of k that will make f(x, y) a legitimate joint probability density function. Use f(x.y) with that value of k as the joint probability density function of X, Y in parts (b),(c).(d),(e (b) Find the probability density functions of X and Y. (c) Find the expected values of X, Y and XY (d) Compute the covariance Cov(X,Y) of X...
4. Let X and Y be continuous random variables with joint density function f(x, y) = { 4x for 0 <x<ys1 otherwise (a) Find the marginal density functions of X and Y, g(x) and h(y), respectively. (b) What are E[X], E[Y], and E[XY]? Find the value of Cov[X, Y]
7. Two random variables X and Y have joint probability density function s(x, y) = $(1 – xy), 0<x< l; 0<y<l. The marginal pdfs for X and Y are respectively S(x) = {(2-x) 0<x< 1; s()= (2-y) 0<y<l. Determine the conditional expectation E(Y|X = x) and hence determine E(Y) [7] (ii) [3] Verify your answer to part (i) by calculating the value of E(Y) directly from the marginal pdf for Y. [Total 10]
Consider random variables X and Y with joint probability density function (Pura s (xy+1) if 0 < x < 2,0 <y S4, fx.x(x, y) = otherwise. These random variables X and Y are used in parts a and b of this problem. a. (8 points) Compute the marginal probability density function (PDF) fx of the random variable X. Make sure to fully specify this function. Explain.
1. a) Let X and Y be random variables with the following joint probability density function (pdf) Зу f(x,y) = 0<y< 2x2,0<x< 1. 2.02 i) Obtain the value for E(Y|X = }). ii) Show the relationship between E[Y|X] and E[XY]. Use this result to obtain E[XY]
2. Suppose X and Y are continuous random variables with joint density function f(x, y) = 1x2 ye-xy for 1 < x < 2 and 0 < y < oo otherwise a. Calculate the (marginal) densities of X and Y. b. Calculate E[X] and E[Y]. c. Calculate Cov(X,Y).
55. Let X and Y be jointly continuous random variables with joint density function fx.y(x,y) be-3y -a < x < 2a, 0) < y < 00, otherwise. Assume that E[XY] = 1/6. (a) Find a and b such that fx,y is a valid joint pdf. You may want to use the fact that du = 1. u 6. и е (b) Find the conditional pdf of X given Y = y where 0 <y < . (c) Find Cov(X,Y). (d)...
Let X and Y be random variables with joint density function F(x,y) O<ysi< otherwise The marginal density of Y is fr() = 3 (1 - ), for 0 < y<1. True False